The fully compressed subgroup membership problem

TL;DR

This paper proves that determining subgroup membership in a free group with compressed representations can be done efficiently in polynomial time, extending Stallings' foldings to handle compressed labels.

Contribution

It introduces a polynomial-time algorithm for the fully compressed subgroup membership problem by adapting Stallings' foldings to compressed labels.

Findings

Polynomial-time solvability of the problem

Extension of Stallings' foldings to compressed labels

Partial answer to Markus Lohrey's question

Abstract

Suppose that $F$ is a free group and $k$ is a natural number. We show that the fully compressed membership problem for $k$-generated subgroups of $F$ is solvable in polynomial time. In order to do this, we adapt the theory of Stallings' foldings to handle edges with compressed labels. This partially answers a question of Markus Lohrey.